Condition REF  is crucial for the convergence of \(\widehat{\beta }_{(s)}\)  and \(\widehat{\beta }_{(s)}^{[-j]}\) . Condition REF  is the same as the condition of Lemma 5.4 in [1]}, and it requires the smoothness of the objective function. Condition REF  holds if there exists a local maximum for every \(s\)  and can be roughly regarded as a minimum-eigenvalue requirement of the “Fisher information" matrix.
When \(J_n\)  and \(S_n\)  are divergent, we need to restrict the eigenvalues of “Fisher information" across all blocks and candidate models, and thus \(\min _{1\le s\le S_n}\)  and \(\min _{1\le j\le J_n}\)  are needed.
